The theory of mathematical turbulence was initiated in the 1930s and 1940s by Taylor, Richardson, Kolmogorov and others, see [1]} and references therein, and has since influenced fluid mechanics, as well as atmospheric sciences and plasma physics heavily. In his sequence of three papers [2]}, [3]}, [4]}, nowadays referred to as K41, Kolmogorov took a probabilistic approach to turbulence and formulated basic hypotheses about fluid flow at high Reynolds numbers and derived predictions based on these. Many of these have later been confirmed by experiments. The idea of studying equations (REF ) in a probabilistic setting
has since been taken up again in many works, in different frameworks, by adding stochastic forcing terms to (REF ), see e.g. [5]}, [6]}, [7]}, or taking uncertain or measure-valued initial data, e.g. [8]}. In the latter case, the solution of (REF ) may not be a function any more but instead a time-parametrized probability measure on the phase space. Global existence of such measure-valued solutions for incompressible flows has been shown in 3D, and even the passage to the limit \(\varepsilon \rightarrow 0\)  can be made rigorous in this case [8]}. However, measure-valued solutions are generally not unique, which can be shown by counterexample even in the case of Burgers' equation [10]}. Hence, measure-valued solutions are too broad a solution concept to resolve the problem of non-uniqueness, and more information or constraints need to be added.
